= Solution
Let $D:\mathcal J\to\mathbf{Fld}$ be connected and nonempty. The limit of the underlying diagram of <commutative ring>[commutative rings] is the subring
$$
L=\left\{(x_j)\in\prod_jD(j):
D(u)(x_i)=x_j\text{ for every }u:i\to j\right\}.
$$
If $x\in L$ is nonzero in one component, it is nonzero in every component: field homomorphisms are injective, and connectedness propagates this fact along zigzags. Hence the componentwise inverses $(x_j^{-1})$ are defined and compatible. Thus $L$ is a field. Since the inclusion $\mathbf{Fld}\hookrightarrow\mathbf{CRng}$ is full, the same cone is limiting in the <category of fields>.
If $\mathcal J$ is disconnected, choose two components and use the constant field $\mathbb F_2$ on one and $\mathbb F_3$ on the other. There is no cone in fields because its apex would map to fields of two different characteristics. Hence $\mathbf{Fld}$ does not have all limits of any disconnected shape.
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